Sometimes when we want to work out the area under a curve, we cannot integrate the function easily. In these cases, we can use numerical integration methods to calculate the area. Numerical methods are not exact methods; rather, they approximate the area as being, for example, a whole lot of rectangles.
One way of working out a fairly good approximate value for the area underneath a curve is to represent the area as lots of thin rectangles. This technique is known as a Riemann Sum.
The general form of a Riemann Sum is:
Say I want to calculate the area beneath the function from x = 0 to x = 5.
Firstly, I must decide how many rectangles I want to use to simulate the area. The more rectangles I use, the more accurate the answer, but the longer it will take. Let’s use 5.
Since the interval is from 0 to 5, and we are using 5 rectangles, each rectangle must be 1 unit wide – Dx = 1.
The mathematical sign with the ‘i=1’ below it is called a sigma sign. All it means in this case is:
· Calculate the value of the function at x = 0 and multiply it by Dx.
· Calculate the value of the function at x = 0 + Dx and multiply it by Dx
· Calculate the value of the function at x = 0 + 2Dx and multiply it by Dx.
· Keep doing this until you get to x = 5 – Dx.
· Add all these results together.
Adding these results together, you get 115 as the area under the curve between x = 0 and x = 5.